The existing theory of discharge breakdown is based on the constant rate paradigm by which the cumulative breakdown probability P(t), i.e., the probability that the breakdown occurred for t′ such that 0≦t′≦t, is given by
      P    ⁡          (      t      )        =      {                                        0            ,                                                                              for                ⁢                                                                  ⁢                t                            ≤                              t                0                                      ,                                                                          1              -                              exp                ⁡                                  (                                      -                                          r                      ⁡                                              (                                                  t                          -                                                      t                            0                                                                          )                                                                              )                                                      ,                                                                              for                ⁢                                                                  ⁢                t                            >                              t                0                                      ,                              with r being a time-independent breakdown rate, and t0 being a minimal breakdown time. The breakdown rate is further expanded as r={dot over (n)}·p where {dot over (n)} is the rate of appearance of eligible electrons at the cathode, while p is the probability that these electrons create an avalanche leading to breakdown. For high intensity discharge (HID) lamps p is of secondary importance because HID lamps are typically overvoltaged to insure ignition, meaning that for all practical purposes p≡1.
The ignition voltage is significantly greater than the operating voltage of the lamp and requires a special igniter circuit that has to be designed and incorporated into the ballast. As the ignition voltage of individual lamps may vary, the circuit must be designed to handle a range of values. A standard igniter uses a brute force approach to start a lamp: a long sequence of ignition pulses is applied in expectation that the lamp will eventually ignite. This approach limits the design parameters for the ballast electronics which supply and manage the ignition voltage and the power used to ignite and then operate the lamp. Lowering the ignition voltage would lead to less expensive and more compact ballasts if it could be achieved while maintaining reliable starting of the lamp.